A Hilbert space proof of the fundamental theorem o

AHilbertSpaceProofoftheFundamentalTheoremofAssetPricinginFiniteDiscreteTimeW.SchachermayerInstitutfurMathematikderUniversitatWien,Strudlhofgasse4,A-1090Wien,Austria.October92Abstract.R.Dalang,A.MortonandW.WillingerhaveprovedabeautifulversionoftheFundamentalTheoremofAssetPricingwhichpertainstothecaseofnitedis-cretetime:Inthiscasetheabsenceofarbitrageopportunitiesalreadycharacterizestheexistenceofanequivalentmartingalemeasure.Thepurposeofthispaperistogiveanelementaryproofofthisimportanttheoremwhichreliesonlyonorthogonalityarguments.Incontrast,theoriginalproofofDalang,MortonandWillingerusesheavyfunctionalanalyticmachinery,inparticularmeasur-ableselectionandmeasure-decompositiontheorems.Wefeelthatthetheorem(aswellasitsproof)shouldbeaccessibletoawiderpublicandwethereforemadeaneorttokeeptheargumentsasselfcontainedaspossible.Inanalchapterwereviewandprovethenecessarytoolsforourpresentationofthetheorem.1.IntroductionWeconsideranRd-valuedstochasticprocess(St)Nt=0whichisindexedbythenitediscretetimesetf0;1;:::;Ng.InmathamaticalnancetheprocessSusuallymodelsthe(discounted)priceprocessofdstocks.TheFundamentalTheoremofAssetPricingstatesthattheexistenceofanequiv-alentmartingalemeasurefortheprocessSisessentiallyequivalenttotheabsenceofarbitrageopportunities.Thetheoremisrightlytermedfundamentalasitallowstorelatetheconceptofpricingbyarbitrage{whichhasexperiencedincreasingim-portancesincetheseminalpapersofF.BlackandM.Scholes[B-S73]andR.Merton[M73]{withthemachineryofmartingaletheory.Inparticularitallowstoreducethepricingofacontingentclaimtocalculatingexpectationvalues,justinthewayactuariesdoforcenturies.The(decisive)dierenceliesonlyinthefactthatonedoesnottaketheexpectationwithrespecttotheoriginalprobabiltymeasurePbutwithrespecttoanarticialrisk-neutralprobabilitymeasureQ,i.e.withrespecttoameasureunderwhichtheprocess(St)Nt=0isamartingale.1980MathematicsSubjectClassication(1991Revision).Primary60H05;.Keywordsandphrases.EquivalentMartingaleMeasure,PricingbyArbitrage.1ThesubtlepointintheFundamentalTheoremofAssetPricingistogiveaprecisemeaningtothewordessentially.Inthegeneralcaseofinniteorcontinuoustimethisproblemturnsouttobeverydelicateandneedsnotionssuchasnofreelunchornofreelunchwithboundedriskgeneralizingtheconceptofnoarbitrageinordertoobtainsaisfactorytheorems(compare[H-K79],[H-P81],[K81],[D-H86],[St90],[A-S90],[F-S90],[D91],[S92]).ButinthepresentlyconsideredcaseofnitediscretetimeDalang,MortonandWillingershowedthatthereisaniceandclear-cuttheoremwhichmaybephrasedbyusingonlytheclassicalnotionofnoarbitrage.Letusgivesomeprecisedenitions:(;F;(Ft)Nt=0;P)willdenotealteredprob-abilityspaceandweassumethattheprocess(St)Nt=0isadaptedtotheltration(Ft)Nt=0.AprobabilitymeasureQonFwillbecalledequivalenttoPifQandPhavethesamenullsetsor{equivalently{ifthemutualRadon-Nikodymderivativesexist.WesaythatanequivalentprobabilitymeasureQisanequivalentmartingalemea-surefor(St)Nt=0if(St)Nt=0isamartingaleunderQ,i.e.,eachStisQ-integrableandforeacht=1;:::;N,wehaveEQ((StSt1)jFt1)0:Wesaythattheprocess(St)Nt=0satisesthenoarbitrageconditioniffort=1;:::;NandeachFt1-measurableboundedRd-valuedfunctionhsuchthat(h(!);St(!)St1(!))0Pa.s.wehave(h(!);St(!)St1(!))=0Pa.s.Here(:;:)denotestheinnerproductonRd.Thenoarbitrageconditionhasadirecteconomicinterpretation:Itshouldnotbepossibletoperformatradingoperationonthestockpriceprocess(St)Nt=0,describedbytherandomvariableh,suchthatthenetresultisalmostsurelynonnegativewithoutbeingalmostsurelyzero.Itisreasonabletoarguethatagoodmodel(St)Nt=0ofanancialmarketshouldsatisfythisassumption.Theargumentisthatotherwisetherewouldbeeconomicagentstakingadvantageofthisarbitrageopportunitywhichwouldquicklymakeitdisappear.ItisalmostobviousthattheexistenceofanequivalentmartingalemeasureQimpliesthattheprocess(St)Nt=0satisesthenoarbitragecondition.Indeed,if(St)Nt=0isamartingalewithrespecttoQthenwehaveforeachFt1-measurableboundedRd-valuedfunctionhthatEQ(h(!);St(!)St1(!))=0:Ifwehaveinadditionthat(h(!);St(!)St1(!))0P-almostsurely(andthere-foreQ-almostsurely)weconcludethat(h(!);St(!)St1(!))=0Q-almostsurely(andthereforeP-almostsurely).Thisshowsthattheprocess(St)Nt=0satisesthenoarbitragecondition.ThepointoftheDalang-Morton-Willingertheoremliesinthefactthatthereverseimplicationalsoholdstrue:21.1Theorem(Dalang,MortonandWillinger).AnadaptedRd-valuedpro-cess(St)Nt=0satisesthenoarbitrageconditionifandonlyifthereexistsanequivalentmartingalemeasure.InthiscasetheequivalentmartingalemeasureQmaybechosensuchthatthedensitydQ=dPisuniformlybounded.Somecommentsonthistheoremareinorder:Ofcourse,thetheoremappliesinparticulartothecased=1,i.e.,theclassicalcasewhereonlyonestockisconsidered.InthiscasethetheoremwasobtainedbyBackandPliska[B-P90]{whoalsoconjec-turedthetheoremforgenerald2N{andtheproofissubstantiallyeasier(compareremark2.8below).Thecased1ismuchmoredelicateandneedssomekindofgeometricargument.Wehavedealtwiththesedicultiesbyusingorthogonalityargumentsinproperlychosenspaces.Oneshouldnotethatthenoarbitrageconditionimposesnointegrab